Entire solutions of a certain type differential-difference equation and differential-difference analogue of Bruck conjecture

In the paper, we find out the precise form of the finite order entire solutions of the following differential-difference equation \[f⁽ᵏ⁾(z)={}{^nⱼ₌₀}{∑} a_j f(z+jc),\] where a₀, a₁,…,aₙ(≠ 0). Also in the paper we study the differential-difference analogue of Brück conjecture and derive a uniqueness result of finite order entire function $f(z)$ having a Borel exceptional small function of $f(z)$, when f⁽ᵏ⁾(z) and ⁿⱼ₌₀∑ aⱼ f(z+jc) share a small function of $f(z)$. The obtained results, significantly generalize and improve the results due to Liu and Dong (Some results related to complex differential-difference equations of certain types, Bull. Korean Math. Soc., 51 (5) (2014), 1453-1467). Some examples are given to ensure the necessity of the condition (s) of our main results.